Tag Archives: word algebra

made in México


maybe, for the presentation \langle a,b,c\mid a^2=1, b^2=1 , c^2=1\rangle, is this the its Cayley’s graph?Nsub3CayleyGcC

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Filed under algebra, group theory, low dimensional topology, math, topology, word algebra

integers assigned respectively


integer assigned respectively

integers assigned respectively

Let me tell that these are words for the group presented as

\langle a,\ b\ \mid \ a^2=e, \ b^2=e\rangle

Este grupo tiene una infinidad de elementos de orden dos: todas las palabras de longitud impar como w=ababa cuando se hace w^2=(ababa)(ababa)=1.

There are  words as w_1=ababab that with the word w_2=bababa they do w_1w_2=w_2w_1=1

Type W_1 words are like w=(ab)^n and W_2 and words as w=(ba)^m, then they form a subgroup H which is isomorphic to \mathbb{Z}. This via (ab)^n\mapsto n, (ba)^n\mapsto -n.

Este subgrupo arma solo dos clases laterales {\mathbb{Z}}_2*{\mathbb{Z}}_2/H=\{eH,\ aH=bH\}. Entonces vemos que

1\to{\mathbb{Z}}\to{\mathbb{Z}}_2*{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1

es exacta.

That is, {\mathbb{Z}}_2*{\mathbb{Z}} is  an extension of \mathbb{Z} by {\mathbb{Z}}_2.

Otra posible extensión es la trivial:

1\to{\mathbb{Z}}\to{\mathbb{Z}}\oplus{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1

It is known that all extensions $E$ in  an exact sequence 1\to{\mathbb{Z}}\to E\to{\mathbb{Z}}_2\to1 are in bijection within the morphisms {\mathbb{Z}}_2\to{\rm out}\mathbb{Z}\cong{\mathbb{Z}}_2

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Filed under algebra, group theory, math

paraphrasis 1


tensors are like any other machines,

they are either a benefit or a hazard,

if they’re a benefit, it’s not my problem…

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Filed under math

supposethathespacebardoesn’texists


that is, writing without spaces between the words… is it a fashion?

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Filed under free group